As transductive learning has not been extensively researched in the deep learning community, we begin by defining the transductive learning problem within the context of deep learning classification. We follow Vapnik's standard formulation from statistical learning theory [1]}. Let \(P(X,Y)\)  be a probability distribution over \(\mathcal {X}\times \mathcal {Y}\) , where \(\mathcal {X}\)  represents an input space (e.g., raw image data), and \(\mathcal {Y}\)  represents a label set corresponding to \(C\)  classes. The learner is provided with a set of labeled instances,
\(S_l \triangleq \lbrace  (x_1, y_1), \ldots , (x_L, y_L) \rbrace \) , where \(x_i \in \mathcal {X}\) , \(y_i \in \mathcal {Y}\) , and a finite set of unlabeled instances, \(X_u \triangleq \lbrace  x_{L+1}, \ldots , x_{L+U} \rbrace \) , where \(x_i \in \mathcal {X}\) .
The objective is to label the unlabeled instances based on this data.
Given a deep neural network \(f_\theta : \mathcal {X}\rightarrow \mathcal {Y}\) , where \(\theta \)  denotes its
parameters, and a loss function \(\ell : \mathcal {Y}\times \mathcal {Y}\rightarrow \mathbb {R}\) , we follow [2]} (Setting 2) and define the (true) risk over the unlabeled set,
\(E_{f_{\theta }}(X_U)\triangleq \frac{1}{U} \sum _{i=1}^U \ell (f_\theta (x_{L+i}),y_{L+i}).\) 
